the graph of f has a horizontal asymptote at y=k
lim x -> +- ∞ f(x) = k
If the graph of a function f has a horizontal asymptote at y=k, it means that as the values of x move towards positive or negative infinity, the values of f(x) approach the value k.
In other words, the line y=k acts as a horizontal boundary for the graph of f. This can be represented mathematically using limits:
lim (x→∞) f(x) = k
lim (x→-∞) f(x) = k
This means that no matter how large or small the x values become, the values of f(x) will get closer and closer to k, but will never actually reach it.
Horizontal asymptotes can occur in different shapes of functions, such as a rational function or an exponential function.
For example, in the graph of the rational function f(x)=2x^2/(x^2+1), the horizontal asymptotes occur when x approaches infinity or negative infinity, and the values of f(x) approach 2. To prove this, we can simplify the rational function by dividing both the numerator and denominator by x^2:
f(x) = 2x^2/(x^2+1) = 2 + (-2/(x^2+1))
As x approaches infinity or negative infinity, the value of (-2/(x^2+1)) approaches zero, and thus the value of f(x) approaches 2.
Therefore, we can conclude that if the graph of f has a horizontal asymptote at y=k, it means that the values of f(x) approach k as x approaches infinity or negative infinity.
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